Local Statistics in Normal Matrix Models with Merging Singularity

TL;DR

This paper analyzes the local correlation kernel of the normal matrix model with merging singularity, revealing anisotropic scaling and universal behaviors related to Painlevé II and elliptic ensembles.

Contribution

It introduces a detailed analysis of local statistics near merging singularities in normal matrix models, utilizing Riemann-Hilbert problems and identifying novel anisotropic scaling behaviors.

Findings

Derived the limiting local correlation kernel at the singularity.

Identified anisotropic scaling behavior with different particle spacing scales.

Observed universal bulk, edge, and sine-kernel statistics in various regimes.

Abstract

We study the normal matrix model, also known as the two-dimensional one-component plasma at a specific temperature, with merging singularity. As the number $n$ of particles tends to infinity we obtain the limiting local correlation kernel at the singularity, which is related to the parametrix of the Painlev\'e~II equation. The two main tools are Riemann-Hilbert problems and the generalized Christoffel-Darboux identity. The correlation kernel exhibits a novel anisotropic scaling behavior, where the corresponding spacing scale of particles is $n^{-1/3}$ in the direction of merging and $n^{-1/2}$ in the perpendicular direction. In the vicinity at different distances to the merging singularity we also observe Ginibre bulk and edge statistics, as well as the sine-kernel and the universality class corresponding to the elliptic ensemble in the weak non-Hermiticity regime for the local correlation function.